1) Quadratic expression and equation
A quadratic expression is a polynomial of degree 2, such as ax² + bx + c, where a ≠ 0. It is called an expression because it has no equality sign.
A quadratic equation is formed when the quadratic expression is set equal to zero:
ax² + bx + c = 0, where a ≠ 0
2) Standard form
Before solving, always rewrite the equation in standard form.
ax² + bx + c = 0
Example: 2x² - 7 = 3x becomes 2x² - 3x - 7 = 0.
3) Sridharacharya’s formula
Sridharacharya’s formula is used to find the roots of a quadratic equation.
x = (-b ± √(b² - 4ac)) / 2a
It gives two roots:
x₁ = (-b + √(b² - 4ac)) / 2a
x₂ = (-b - √(b² - 4ac)) / 2a
4) Derivation of the formula
Start with the standard quadratic equation:
ax² + bx + c = 0
Divide by a:
x² + (b/a)x + c/a = 0
Move the constant term to the other side:
x² + (b/a)x = -c/a
Add (b/2a)² to both sides:
x² + (b/a)x + (b/2a)² = -c/a + (b/2a)²
The left side becomes a perfect square:
(x + b/2a)² = (b² - 4ac) / 4a²
Taking square roots:
x + b/2a = ± √(b² - 4ac) / 2a
x = (-b ± √(b² - 4ac)) / 2a
5) Discriminant
The expression D = b² - 4ac is called the discriminant. It tells us the nature of the roots.
D = b² - 4ac
6) Nature of roots
| Discriminant | Nature of roots |
|---|---|
| D > 0 | Two real and distinct roots |
| D = 0 | Two real and equal roots |
| D < 0 | No real roots |
7) Graph of quadratic equation
The graph of a quadratic equation is a parabola.
- If a > 0, the graph opens upward.
- If a < 0, the graph opens downward.
8) When the quadratic is positive or negative
Let f(x) = ax² + bx + c and let its real roots be α and β, where α < β.
| Condition | Sign of f(x) |
|---|---|
| a > 0, x < α or x > β | Positive |
| a > 0, α < x < β | Negative |
| a < 0, x < α or x > β | Negative |
| a < 0, α < x < β | Positive |
9) How to solve quadratic equations
By factorization
Split the middle term or write the quadratic as a product of two linear factors.
x² + 5x + 6 = 0
(x + 2)(x + 3) = 0
x = -2, -3
By Sridharacharya’s formula
Use the formula when factorization is difficult.
x = (-b ± √(b² - 4ac)) / 2a
By completing the square
Convert the equation into a perfect square and solve. This method is also used in the derivation of the quadratic formula.
10) Relations between roots and coefficients
If α and β are the roots of ax² + bx + c = 0, then:
α + β = -b/a
αβ = c/a
Special case when α = β
If the roots are equal, then the discriminant is zero:
D = 0
Repeated root = -b / 2a
Also, when α = β, then α + β = 2α and αβ = α².
11) Quick revision
- Quadratic equation: ax² + bx + c = 0, a ≠ 0
- Formula: x = (-b ± √(b² - 4ac)) / 2a
- Discriminant: D = b² - 4ac
- D > 0 → real and distinct roots
- D = 0 → real and equal roots
- D < 0 → no real roots
- a > 0 → parabola opens upward
- a < 0 → parabola opens downward
- α + β = -b/a
- αβ = c/a